Gap sequences and Topological properties of Bedford–McMullen sets*The research of Miao is partially supported by Shanghai Key Laboratory of PMMP (18dz2271000) and NSFC Grant 12071137. The research of Ruan is partially supported by NSFC Grant 11771391, ZJNSF Grant LY22A010023 and the Fundamental Research Funds for the Central Universities of China Grant 2021FZZX001-01. (4th August 2022) - Liang, Zhen; Miao, Jun Jie; Ruan, Huo-Jun
Abstract: In this paper, we study the topological properties and the gap sequences of Bedford–McMullen sets. First, we introduce a topological condition, the component separation condition (CSC), and a geometric condition, the exponential rate condition (ERC). Then we prove that the CSC implies the ERC, and that both of them are sufficient conditions for obtaining the asymptotic estimate of gap sequences. We also explore topological properties of Bedford–McMullen sets and prove that all normal Bedford–McMullen sets with infinitely many connected components satisfy the CSC, from which we obtain the asymptotic estimate of the gap sequences of Bedford–McMullen sets without any restrictions. Finally, we apply our result to Lipschitz equivalence.